Peter McMullen

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Peter McMullen
Born (1942-05-11) 11 May 1942 (age 80)
Alma mater Trinity College, Cambridge
Known for Upper bound theorem, McMullen problem
Scientific career
Fields Discrete geometry
Institutions Western Washington University (1968–1969)
University College London

Peter McMullen (born 11 May 1942) [1] is a British mathematician, a professor emeritus of mathematics at University College London. [2]


Education and career

McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at the University of Birmingham, where he received his doctorate in 1968. [3] and taught at Western Washington University from 1968 to 1969. [4] In 1978 he earned his Doctor of Science at University College London where he still works as a professor emeritus. In 2006 he was accepted as a corresponding member of the Austrian Academy of Sciences. [5]


McMullen is known for his work in polyhedral combinatorics and discrete geometry, and in particular for proving what was then called the upper bound conjecture and now is the upper bound theorem. This result states that cyclic polytopes have the maximum possible number of faces among all polytopes with a given dimension and number of vertices. [6] McMullen also formulated the g-conjecture, later the g-theorem of Louis Billera, Carl W. Lee, and Richard P. Stanley, characterizing the f-vectors of simplicial spheres. [7]

The McMullen problem is an unsolved question in discrete geometry named after McMullen, concerning the number of points in general position for which a projective transformation into convex position can be guaranteed to exist. It was credited to a private communication from McMullen in a 1972 paper by David G. Larman. [8]

He is also known for his 1960s drawing, by hand, of a 2-dimensional representation of the Gosset polytope 421, the vertices of which form the vectors of the E8 root system. [9]

Awards and honours

McMullen was invited to speak at the 1974 International Congress of Mathematicians in Vancouver; his contribution there had the title Metrical and combinatorial properties of convex polytopes. [10]

He was elected as a foreign member of the Austrian Academy of Sciences in 2006. [11] In 2012 he became an inaugural fellow of the American Mathematical Society. [12]

Selected publications

Research papers
Survey articles

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  1. Peter McMullen, Peter M. Gruber, retrieved 2013-11-03.
  2. UCL IRIS information system, accessed 2013-11-05.
  3. McMullen, Peter; Schulte, Egon (12 December 2002), Abstract and regular polytopes, ISBN   9780521814966 , retrieved 2022-05-11
  4. Peter McMullen Collection, 1967-1968, Special Collections, Wilson Library, Western Washington University, retrieved from 2013-11-03.
  5. "Austrian Academy of Sciences: Peter McMullen" . Retrieved 2022-05-11.
  6. Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer, p. 254, ISBN   9780387943657, Finally, in 1970 McMullen gave a complete proof of the upper-bound conjecture – since then it has been known as the upper bound theorem. McMullen's proof is amazingly simple and elegant, combining to key tools: shellability and h-vectors.
  7. Gruber, Peter M. (2007), Convex and discrete geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Berlin: Springer, p. 265, ISBN   978-3-540-71132-2, MR   2335496, The problem of characterizing the f-vectors of convex polytopes is ... far from a solution, but there are important contributions towards it. For simplicial convex polytopes a characterization was proposed by McMullen in the form of his celebrated g-conjecture. The g-conjecture was proved by Billera and Lee and Stanley.
  8. Larman, D. G. (1972), "On sets projectively equivalent to the vertices of a convex polytope", The Bulletin of the London Mathematical Society, 4: 6–12, doi:10.1112/blms/4.1.6, MR   0307040
  9. "A picture of the E8 root system". American Institute of Mathematics. Retrieved 2022-05-11.
  10. ICM 1974 proceedings Archived 2017-12-04 at the Wayback Machine .
  11. Awards, Appointments, Elections & Honours, University College London, June 2006, retrieved 2013-11-03.
  12. List of AMS fellows, retrieved 2013-11-03.